Bidding in Online Auctions

ABSTRACT

Embodiments are directed to systems, methods, and apparatus for bidding in online auctions. In one embodiment, bids for advertising include an amount that is a function of an expected value-per-click and a fraction of a budget already spent for advertising slots.

BACKGROUND

Online search engines provide a popular tool for searching keywords overthe internet. Search engines and corresponding online auctions globallygenerate billions of dollars a year in revenue. The results page of akeyword search is therefore an effective place for advertisers to reachan engaged audience.

Using an automated auction mechanism, search engines sell the right toplace ads next to keyword results and alleviate the auctioneer from theburden of pricing and placing ads. The intent of the consumer is matchedwith that of the advertiser through an efficient cost/benefit enginethat favors advertisers who offer that which consumers seek.

On the advertising side, companies spend billions of dollars each yearin marketing with an increasingly large portion of that money dedicatedto search engine marketing. Since such large sums of money are beingspent, advertisers strive to strategically bid against competingadvertisers while maximizing return for advertising dollars.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an exemplary data processing network in accordancewith an exemplary embodiment.

FIG. 2 illustrates an exemplary search engine and bid optimizationengine in accordance with an exemplary embodiment.

FIG. 3 illustrates an exemplary flow diagram in accordance with anexemplary embodiment.

DETAILED DESCRIPTION

Exemplary embodiments are directed to systems, methods, and apparatusfor budget constrained bidding in online keyword auctions. Exemplaryembodiments optimize bids for advertisings bidding in a competitiveenvironment for advertising slots in an online auction.

One embodiment is directed to sponsored search auctions hosted by searchengines that allow advertisers to select relevant keywords, allocatebudgets to those terms, and bid on different advertising positions foreach keyword in a real-time auction against other advertisers. Exemplaryembodiments provide optimal bid management of advertising budgets,especially for large advertisers who need to manage thousands ofkeywords and spend tens of millions on such advertising.

In one embodiment, optimization of bid management is cast as an online(multiple choice) knapsack problem and corresponding algorithms for theonline knapsack problem achieve a provably optimal competitive ratio.This allows for the optimization of the bidding process, whileoptimizing bids to best achieve the goals of the program. To maximizerevenue from sponsored search advertising, the bidding strategy can beoblivious (i.e., without knowledge) of other bidder's prices and/orclick-through-rates for those positions. Further, bidding algorithms inaccordance with exemplary embodiments are evaluated using both syntheticdata and real bidding data obtained from online websites.

FIG. 1 illustrates an exemplary system or data processing network 10 inwhich exemplary embodiments are practiced. The data processing networkincludes a plurality of computing devices 20 in communication with anetwork 30 that is in communication with one or more computer systems orservers 40.

For convenience of illustration, only a few computing devices 20 areillustrated. The computing devices include a processor 12, memory 14,and bus 16 interconnecting various components. Exemplary embodiments arenot limited to any particular type of computing device or server sincevarious portable and non-portable computers and/or electronic devicesmay be utilized. Exemplary computing devices include, but are notlimited to, computers (portable and non-portable), laptops, notebooks,personal digital assistants (PDAs), tablet PCs, handheld and palm topelectronic devices, compact disc players, portable digital video diskplayers, radios, cellular communication devices (such as cellulartelephones), televisions, and other electronic devices and systemswhether such devices and systems are portable or non-portable.

The network 30 is not limited to any particular type of network ornetworks. The network 30, for example, can include one or more of alocal area network (LAN), a wide area network (WAN), and/or the internetor intranet, to name a few examples. Further, the computer system 40 isnot limited to any particular type of computer or computer system. Thecomputer system 40 may include personal computers, mainframe computers,servers, gateway computers, and application servers, to name a fewexamples.

Those skilled in the art will appreciate that the computing devices 20and computer system 40 connect to each other and/or the network 30 withvarious configurations. Examples of these configurations include, butare not limited to, wireline connections or wireless connectionsutilizing various media such as modems, cable connections, telephonelines, DSL, satellite, LAN cards, and cellular modems, just to name afew examples. Further, the connections can employ various protocolsknown to those skilled in the art, such as the Transmission ControlProtocol/Internet Protocol (“TCP/IP”) over a number of alternativeconnection media, such as cellular phone, radio frequency networks,satellite networks, etc. or UDP (User Datagram Protocol) over IP, FrameRelay, ISDN (Integrated Services Digital Network), PSTN (Public SwitchedTelephone Network), just to name a few examples. Many other types ofdigital communication networks are also applicable. Such networksinclude, but are not limited to, a digital telephony network, a digitaltelevision network, or a digital cable network, to name a few examples.Further yet, although FIG. 1 shows one exemplary data processingnetwork, exemplary embodiments can utilize various computer/networkarchitectures.

For convenience of illustration, an exemplary embodiment is illustratedin conjunction with a search engine. This illustration, however, is notmeant to limit embodiments with search engines. Further, exemplaryembodiments do not require a specific search engine. The search enginecan be any kind of search engine now known or later developed. Forexample, exemplary embodiments are used in conjunction with existingsearch engines (such as PageRank and variations thereof) or searchengines developed in the future.

FIG. 2 illustrates an exemplary system 200 that includes a search engine202 and bid optimization engine 204. As one example, the search engine202 and bid optimization engine 204 are programs stored in the memory ofcomputer system 40. The search engine enables a user to requestinformation or media content having specific criteria. The request, forexample, can be entered as keywords or a query. Upon receiving thequery, the search engine 202 retrieves documents, files, or informationrelevant to the query. The bid optimization engine 204 optimizes bidsfor advertising slots when the search results are displayed to a user.

For simplicity of illustration, the search engine 202 includes a webcrawler 210, a search manager 220, and a ranking algorithm 230 coupledto one or more processors 245 and a database 240. The bid optimizationengine 204 includes a bid optimizing algorithm 260 coupled to one ormore processors 270. The search engine 202 and bid optimization engine204 are discussed in connection with the flow diagram 300 of FIG. 3.

According to block 310, the web crawler 210 crawls or searches thenetwork and builds an associated database 240. The web crawler 210 is aprogram that browses or crawls networks, such as the internet, in amethodical and automated manner in order to collect or retrieve data forstorage. For example, the web crawler can keep a copy of all visited webpages and indexes and retain information from the pages. Thisinformation is stored in the database 240. Typically, the web crawlertraverses from link to link (i.e., visits uniform resource locators,URLs) to gather information and identify hyperlinks in web pages forsuccessive crawling.

One skilled in the art will appreciate that numerous techniques can beused to crawl a network, and exemplary embodiments are not limited toany particular web crawler or any particular technique. As one example,when web pages are encountered, the code comprising each web page (e.g.,HyperText Markup Language or HTML code) is parsed to record its linksand other page information (example, words, title, description, etc.). Alisting is constructed containing an identifier (example, web pageidentifier) for all links of a web page. Each link is associated with aparticular identifier. The listing is sorted using techniques known inthe art to distinguish the web pages and respective links. Therelationship of links to the parsed web pages and the order of the linkswithin a web site are maintained. After sufficient web sites have beencrawled, the recorded or retrieved information is stored in the database240.

Once the database 240 is created, the search engine 202 can processsearch queries and provide search results. One skilled in the art willappreciate that numerous techniques can be used to process searchqueries and provide search results, and exemplary embodiments can beutilized with various techniques.

According to block 320, the bid optimization engine 204 receivesinformation from an advertiser concerning the placement of ads foronline auctions. By way of example, this information includes, but isnot limited to, one or more of keywords, a budget, and a time period forutilizing the budget.

By way of example, suppose there are N+1 bidders {0, . . . ,N}interested in a single keyword. Bidder 0 is the default advertiser, andhe wants to maximize his profit over a period of time T. Let V denotethe expected value-per-click for the default advertiser, and he has abudget of B over time period T (e.g. if T is 24 hours, B is the dailybudget). Here the budget constraints is a hard constraint, in the sensethat once exhausted, it cannot be refilled; budget remaining at the endof the period T is taken away. Once a bidder exhausts his budget, heleaves the auction.

According to block 330, the search manager 220 receives a query (such askeywords) from a user or computing device (such as computing device 20in FIG. 1). The search manager 220 can perform a multitude of differentfunctions depending on the architecture of the search engine. By way ofexample and not to limit exemplary embodiments, the search manager 220tracks user sessions, stores state and session information, receives andresponds to search queries, and coordinates the web crawler and rankingalgorithm, to name a few examples.

According to block 340, the search engine retrieves and ranks the searchquery. By way of example, the search engine 202 accesses the database240 to find or retrieve information that correlates to the query. As anexample, the search manager 220 could retrieve from the database 240 allweb sites that have a title and description matching keywords in thequery. The search manager 220 then initiates the ranking algorithm 230to score and rank the information (for example, the retrieved web sites)retrieved from the database 240.

According to block 350, the bid optimization, engine 204 optimizes bidson advertising positions against other advertisers. Generally, for eachkeyword and each time period, exemplary embodiments determine how muchmoney an advertiser should bid to obtain a slot or advertising positionon the search results page in order to maximize return on investment(ROI).

In one embodiment, for each user click on its ad, the advertiser obtainsrevenue that is the expected valise-per-click and a profit that is equalto the difference between revenue and cost. The advertiser (or the agenton behalf of the advertiser) has a budget constraint and would like tomaximize either the revenue or the profit. These budget constraintsarise out of the ordinary operational constraints of the firm and itsinteractions with its partners, as well as being a generic feature ofkeyword auction services themselves.

One embodiment uses competitive analysis to evaluate bidding strategiesand compares results with the maximum profit attainable by theomniscient bidder who knows the bids of all the other users ahead oftime. This competitive analysis framework has been used in theworst-case analysis of online algorithms and helps to convert theproblem of devising bidding strategies to designing algorithms foronline knapsack problems. The most general online knapsack problemadmits no online algorithms with any non-trivial competitive ratio, theauction scenario suggests a few constraining assumptions that enableexemplary embodiments to provide optimal online algorithms.

The bidding strategies suggested by the online algorithms and inparticular the bidding strategy for revenue maximization can be statedas follows:

At any time t, if the fraction of budget spent is z(t), bid V/ψ(z(t)).

Here V is the expected value-per-click of the keyword, and ψ(z) is acontinuous function of z specified later. Thus the bidding price dependsonly on the value of the keyword and the fraction of budget spent. Thestrategy is oblivious in the sense that it does not need to considerother players' bids or how frequently queries arrive.

According to block 360, a determination is made of the results of thebid from advertisers and slots are allocated to the bidders. By way ofexample, the bidding strategies in accordance with exemplary embodimentsare based on the current policy used by search engines to display theirads. For instance, embodiments assume that at each query of a keyword,the highest bidder gets first position, the second highest gets thesecond position and so on. Moreover, the pricing scheme is thegeneralized second price scheme where the advertiser in the i-thposition pays the bid of the (i+1)-th advertiser whenever the former'sad is clicked on.

In one embodiment, bidders bid on the keyword, and are allowed to changetheir bids at any moment of time. One assumption is that the bids arevery small compared to the budget of Bidder 0. As soon as a query forthe keywords arrives, the search engine allocates S slots to bidders asfollows: It takes the S highest bids, b₁≧b₂≧ . . . ≧b_(S) and displayss-th bidder's ad in slot s. Moreover, if any user clicks on the ad atthe s-th slot, the search engine charges the s-th bidder a priceb_(s+1), if s<S or a minimum fee b_(min) (for example, 10¢). Hence, itcan be assumed that all the bids are at least b_(min).

Each slot s has a click-through rate α(s), which is defined as theexpected number of clicks on an ad divided by the total number ofimpressions (displays). Usually α(s) is a decreasing function of s. Eachtime his ad in slot s is clicked. Bidder 0 gets a profit of V−b_(s+1)where b_(s+1) is the bid of the advertiser in the (s+1)-th slot orb_(min) if s=S. Suppose the time interval T is discretized into periods{1,2, . . . , T}, such that, within a single time period t, no bidderchanges his bid. Let X(t) denote the expected number of queries for thekeyword in time period t. Moreover, suppose Bidder 0 can make his bid intime period t after seeing all other bidders' bids. This assumption doesnot matter much and is mainly for explanation purposes. The problemfaced by Bidder 0 is to decide, how much to bid at each time period t inorder to maximize its profit while keeping its total cost within itsbudget.

According to block 370, the ranked information is then displayed to theuser or provided to the computing device. Further, the ads are displayedwith the search results according to the bid results. The information isdisplayed, for example, in a hierarchical format with the most relevantinformation (for example, webpage with the highest score) presentedfirst and the least relevant information (for example, webpage with thelowest score) presented last. The ads are displayed according to thewinning bids (i.e., the ad with the highest bid being displayed first,the ad with the next highest bid being displayed second, etc.).

If a modified or new search is requested, according to block 380, thenthe flow diagram loops back to block 330; otherwise, the flow diagramwaits for new search requests 390.

Exemplary embodiments are further described below with headings providedfor various sections.

Bidding Strategies and Knapsack Problems

If bids of all the agents are known at each time period, then the bestbidding strategy corresponds to solving an offline knapsack problem. Toillustrate this concept, discussion begins with a single-slot case wherethere is only one ad slot. At each time period t, let b(t) be themaximum bid on the keyword among bidders 1 to N. The omniscient bidderknows all the bids {b(t)}^(T) _(t=1.) To maximize his profit, theomniscient bidder should bid higher that b(t) at those time periodswhich give him maximum profit and keep his total cost within budget.Winning at time t costs him w(t)=b(t) X(f)α and earns him profitπ(t)=(V=b(t))X(t)α, where X(t)α is the number of clicks at time periodt. Thus, the omniscient bidder should choose time periods S ⊂ T tomaximize π(S)=Σ_(t-CS) π(t) satisfying the constraint w(S)=Σ_(t-CS)w(t)≦B. This is a standard instance of the classic 0/1 knapsack problem,which is defined as following: Given a knapsack of capacity B and Titems of profit and weight (π(t), w(t)) for 1≦t≦T, select a subset ofitems to maximize the total profit with total weight of selected itemsbounded by B. For the case of maximizing revenue, it is similar exceptthat π(t)=VX(t)α for each item t. However, for the keyword auctionproblem, items arrive in an online fashion. At each period t, Bidder 0has to make a decision of either overbidding b(t) or not. Bidder 0 doesnot know the future, and furthermore, it could neither recall timeinstances gone nor revoke its decision of outbidding later. Thusdesigning a bidding strategy corresponds to designing an algorithm forthe online knapsack problem.

The case of multiple slots is captured by the online version of avariant of the classical knapsack problem, the multiple-choice knapsackproblem. This topic is more fully discussed below in the sectionentitled Multiple-Slot Auctions and Online MCKP.

The knapsack problem is a classical problem in operations research andtheoretical computer science. For this discussion, two reasonableassumptions are made on the items of the knapsack, which allowdevelopment of interesting online algorithms. The assumptions are statedbelow and justified in the section entitled Single-Slot Auctions andOnline Knapsack Problems.

The assumptions are:

-   -   1. Each item has weight much smaller than the capacity of the        knapsack, that is, w(t)<<B for each item t.    -   2. The value-to-weight ratio of each item is both lower and        upper bounded, i.e., L≦π(t)/w(t)≦U, ∀t.

Results

In the case of single-slot auctions, the bidding strategy corresponds toonline algorithms for the classical 0/1 knapsack problem. One embodimentis an algorithm for the online knapsack problem with competitive ratioln(U/L)+1, and also a matching lower bound of ln(U/L)+1. Therefore anexemplary embodiment algorithm is provably optimal in the worst casesense. The online knapsack algorithm is translated into a biddingstrategy for the single-slot auction. These strategies are oblivious,and thus work even if other bidders' bids were not known. It alsoimplies that the strategy is an approximate dominant strategy in thesense that it is an approximate best response to any bid profile ofother bidders.

As discussed below, results are extended to the case of multiple-slotauctions. One embodiment gives a (ln(U/L)+2)-competitive onlinealgorithm for a variant of the classical knapsack problem called themultiple-choice knapsack problem (MCKP). The algorithm is translated forOnline-MCKP to bidding strategies for the multiple-slot case in order toobtain both profit-maximizing and revenue-maximizing bidding strategies.The profit maximizing strategy is not oblivious and requires knowledgeof other players' bids and also the click-through-rates of all slots.The revenue-maximizing strategy remains oblivious.

The reason why the multiple-slot profit-maximizing strategy turnsnon-oblivious is subtle: It might be more profitable for an advertiserto appear in a less desirable (lower) slot and pay less than appearingin a higher slot which gives more clicks.

For ease of explanation, discussion is restricted to a single keyword.Exemplary embodiments, though, extend to the general case of multiplekeywords and multiple slots per keyword, with V replaced by V_(max), themaximum valuation-per-click among all the keywords.

One embodiment implements these bidding strategies and evaluates themusing both synthetic bidding data and real bidding data obtained from anonline website. One embodiment modifies the strategy by adding a snipingheuristic, which while maintaining the same theoretical bounds, performsmuch better empirically. One embodiment also utilizes parameter tuningfor the performance of bidding algorithms.

Single-Slot Auctions and Online Knapsack Problems

In this section, an embodiment focuses on single-slot auctions and thecorresponding online knapsack problem. One embodiment utilizesalgorithms for the online knapsack problem and translates them back intobidding strategies for single-slot keyword auctions. Before presentingthe algorithms, an explanation of the previous assumptions is provided.

Recall that the unique item at time period t has a weight w(t) and aprofit π(t) where:

w(t)≡b(t)X(t)α, π(t)≡(V−b(t))X(t)α.

For revenue maximization, π(t) corresponds to the revenue of winning thebid, thus π(t)=VX(t)α. Here X(t)α is the number of expected clicks onthe displayed ad in time period t. The first assumption of w(t)<<Bfollows since the budget of the agent is usually much larger than themoney spent in small time periods as the bids are small. For the secondassumption, separation is made into two cases. In the case of profitmaximization, since π(t)/w(t)=V/b(t)−1, it suffices to setU=V/b_(min)−1. To get a lower bound on the profit-to-weight ratio, it isnoted that if b(t) is close to V, then not too much is lost by notbidding at those time intervals. Specifically, if a bid is made of onlywhen b(t)≦V/(1+ε) for some fixed ε>0, the maximum amount of profit lostfrom not bidding in these time periods is bounded by εB. If ε is small,then the profit loss can be negligible. In other words, one embodimentsets L=ε and ignores all items with efficiency smaller than ε. Thisresults in a maximum profit loss of εB, and it will not affect much thealgorithm performance if the total profit of the algorithm is relativelylarge. In the case of revenue maximization, then π(t)/w(t)=V/b(t). Hereit suffices to set U≡V/b_(min). For the lower bound with revenuemaximization, it is reasonable to assume that the optimum strategy wouldnot bid when b(t) is higher than V. This holds when there are enoughitems with value-to-cost ratio at least 1 to consume the entire budget.Otherwise, the optimal solution needs to take items which cost more thantheir value, and the budget seems unnecessarily large. Therefore,assuming that the optimum never bids when b(t) is higher than V, we onlyneed to consider items with efficiency at least 1, i.e., set L=1.

The Online Knapsack Problem

Given an online algorithm A, we say that A is c-competitive (has acompetitive ratio of c) if for any input sequence of items σ, we haveOPT(σ)≦c A (σ), where A(σ) is the profit of A given σ, and OPT(σ) is themaximum profit obtained by any offline algorithm with the knowledge ofσ.

For all items t, w(t)<<B and L≦π(t)/w(t)≦U. Given the lower bound L andupper bound U for item efficiency, it is verifiable that the algorithmwhich keeps selecting items until the knapsack is full gives acompetitive ratio of U/L. Next the algorithm for the online knapsackproblem is presented, which achieves an optimal competitive ratio boundln(U/L)+1. In the remainder of the discussion, e denotes the base of thenatural logarithm,

Algorithm: Online KP Threshold

Let Ψ(z)≡(Ue/L)^(z)(L/e). At time t, let z(t) be the fraction ofcapacity filled, pick element t iff:

π(t)/w(t)≧Ψ(z(t)).

Observe that for z ∈[0, z], where z≡1/ln(Ue/L), Ψ(z)≦L, thus thealgorithm will pick all items available until z fraction of the knapsackis filled, When z=1, Ψ(z)=U, and since Ψ is strictly increasing, thealgorithm will not spend more than its budget.

The above algorithm uses just one threshold function to select items,and the threshold function is a specialized exponential function of itscapacity filled.

As a theorem, for any input sequence σ, if A(σ) is the profit obtainedby ONLINE-KP-THRESHOLD and OPT (σ) is the maximum profit that can beattained, then:

OPT(π)≦A(σ)(ln(U/L)+1).

In other words, the above algorithm has a competitive ratio ofln(U/L)+1.

Bidding Strategies for Single-Slot Auctions

We now construct the bidding strategies suggested by the algorithmONLINE-KP-THRESHOLD. Consider the profit-maximizing case first. Anassumption is made that b(t)≦V/(1+ε) for some ε. Set U=V/b_(min)−1 andL=ε. At time t, suppose Bidder 0 spent z(t) fraction of its budget. Thiscorresponds to the fact that z(t) fraction of the knapsack is filled. Asper the algorithm, Bidder 0 must win the bid if and only if theefficiency of the next bid is at least Ψ(z(t)). That is, Bidder 0 mustbid higher than b(t) iff(V−b(t))/b(t)≧Ψ(z(t)) or equivalently,b(t)≦V/(1+Ψ(z(t))), which means bidding V/(1+Ψ(z(t))) suffices. Thebidding strategy is stated below:

Bidding Strategy: PROFIT-MAXIMIZING SINGLE-SLOT

Fix ε>0. Let Ψ(z)≡(Ue/ε)^(z) (ε/e).

At time t, if fraction of budget spent is z(t), then bid

b ₀(t)=V/(1+Σ(z(t))).

The above bidding strategy for profit-maximization single-slot auctionshas the following performance guarantee: For any fixed ε>0, let Profitdenote the profit obtained by the profit-maximizing bidding strategy,then

OPT≦εB+ln(e(V−b _(min))/(ε b _(min)))·Profit

where b_(min) is the minimum bid required of the system and OPT is themaximum profit obtained by the omniscient bidder.

Similarly, using U=V/b_(min) and L=1, the following revenue-maximizingbidding strategy is obtained:

Bidding Strategy: REVENUE-MAXIMIZING SINGLE-SLOT

At time t, if fraction of budget spent is z(t), then bid

b ₀(t)=V/Ψ(z(t)).

Here, let Revenue be the revenue obtained by the revenue-maximizingbidding strategy and OPT be the maximum revenue obtained by theomniscient bidder. Assuming that OPT does not contain any item t withb(t)>V, then

OPT≦ln(eV/b _(min))·Revenue

where b_(min) is the minimum bid requirement of the auction system.

Multiple-Slot Auctions and Online MCKP

Exemplary embodiments extend to the case of multiple slots. The strategyin the multiple-slot case corresponds to the online multiple-choiceknapsack problem. The multiple-choice knapsack problem (MCKP) is thegeneralization of the knapsack problem: Given a knapsack of capacity B,and T sets of items N₁, N₂, . . . N_(T), the goal is to choose at mostone item from each set to maximize value and not exceed the knapsackcapacity. In the online version, the sets come online and we need tomake our decision on the fly. Items once picked cannot be disposed, andsets cannot be recalled.

Next it is shown how to model the multiple-slot keyword auction probleminto Online MCKP. Once again, the budget corresponds to the capacity ofthe knapsack. At each time period t, let b₁(t)≧b₂(t)≧ . . . ≧b_(S)(t) bethe S highest bids. To win slot s, bidder 0 needs to bid b_(s)(t). Thisincurs a cost w_(s)(t) and a profit π_(S)(t) where

w _(s)(t)≡b _(s)(t)X(t)α(s), π_(S)(t)≡(V−b _(s)(t))X(t)α(s),

and α(s) is the click-through-rate of slot s. The S slots at each timeperiod correspond to the set arriving at time t. Since Bidder 0 can winat most one slot at each time period, the omniscient bidder needs tosolve the multiple-choice knapsack problem while Bidder 0 needs to solvethe online multiple-choice knapsack problem. Once again, assume that (1)w_(s)(t)<<B and (2) L≦π_(s)(t)/w_(s)(t)≦U for all s, t.

Algorithm for Online MCKP

The algorithm for online-MCKP is similar to that for Online-KP, which isstated below:

Algorithm ONLINE-MCKP-THRESHOLD

Let Ψ(z)≡(Ue/L)^(z)(L/e),

At time t, let z(t) denote the fraction of capacity filled,

$E_{t} = \left\{ {s \in N_{1}} \middle| {\frac{\pi_{s}(t)}{w_{s}(t)} \geq {\Psi \left( {z(t)} \right)}} \right\}$

pick element s ∈E₁ with maximum π_(s)(t).

This algorithm has a competitive ratio of ln(U/L)+2.

Bidding Strategies for Multiple-Slot Auctions

For multiple-slot auctions, consideration is given to theprofit-maximizing case first. As in the single-slot case, we onlyconsider slots with b_(s)(t)≦V/(1+ε) for all s,t. This implies that theefficiency of each bid is upper bounded by U=V/b_(min)−1 and lowerbounded by ε. So the following bidding strategy is obtained:

Bidding Strategy PROFIT-MAXIMIZING MULTIPLE-SLOT

Fix ε>0. Let Ψ(z)≡(Ue/ε)^(z)(ε/e).

At time t, let z(t) be fraction of budget spent,

E _(t) ≡{s|b _(s)(t)≦V/(1+Ψ(z(t)))},

bid b_(s)(t) where s=arg max_(s∈Et)(V−b_(s)(t))α(s).

Note that the bidding strategy is still oblivious of X(t), however itnow requires knowing the bids b_(s)(t) and also α(s). The above biddingstrategy has a performance guarantee, stated as the following theorem:Let Profit denote the profit obtained by our profit-maximizing biddingstrategy. Then for any ε>0,

OPT≦εB+(ln(V/ε b _(min))+2)Profit

where b_(min) is the minimum bid required of the system and OPT is themaximum profit obtained by the omniscient bidder.

For revenue maximization, the bidding strategy is similar to profitmaximization except that it can actually find the slot s in time t tomaximize the revenue. This is because, the revenue obtained on biddingb_(s)(t) is VX(t)α(s). Given that α(s) is a decreasing function,maximizing VX(t)α(s) is equivalent to minimizing s, i.e., to find therank s as low as possible. Since the efficiency condition imposes thatthe winning slot has b_(s)(t)≦V/Ψ(z(t)), the bid should be equal tothis. Thus a bidding strategy exists for revenue-maximizingmultiple-slot auctions which are the same as that for single-slotauctions, which has the desirable property of obliviousness.

Strategy Modification

Embodiments can be modified to have improved empirical performance. As anegative, the strategy (which is described in the following) does notremain oblivious any more: it requires knowledge of X(t), the trafficfunction and also α, the click through-rate of the slot.

A weakness can occur if the strategy is unaware of the time remaining inthe auction. It stops overbidding too early, missing out possibleadvantageous bids later on. Thus a potential performance improvement issnipping towards the end of the auction. At time t, suppose the fractionof budget remaining is y(t)=1−z(t). Moreover assume future click trafficX(τ)α for τ>t is known. With this assumption, a modified bid strategyis:

Bidding Strategy: PROFIT-MAXIMIZING SINGLE-SLOT WITH SNIPING

Fix tCS>0. Let Ψ(z)≡(Ue/tCS)^(z)(tCS/e).

At time t, if fraction of budget spent is z(t), bid

$\max {\left\{ {\frac{V}{1 + {\Psi \left( {z(t)} \right)}},{\min \left\{ {V,\frac{\left( {1 - {z(t)}} \right)B}{\int_{t}^{T}{{X(\tau)}\alpha {\tau}}}} \right\}}} \right\}.}$

In one embodiment, this strategy performs better than the originalstrategy, although it requires the knowledge of X(t) and α.

The above sniping heuristic can be generalized to the multiple-slot caseand it is formally described below:

Bidding Strategy: Multiple-Slot with Sniping.

At time t, let z(t) denote fraction of budget spent ρ=Ψz(t)).

For each slot s,

${{{if}\mspace{14mu} {b_{s}(t)}} \leq \frac{\left( {1 - {z(t)}} \right)B}{{\alpha (s)}{\int_{t}^{T}{{X(\tau)}{\tau}}}}},{\rho = {\min \left\{ {\rho,\frac{\pi_{s}(t)}{w_{s}(t)}} \right\}}}$$E_{t} = \left\{ s \middle| {\frac{\pi_{s}(t)}{w_{s}(t)} \geq \rho} \right\}$

bid b_(s)(t) where s ∈ E, with max π_(s)(t).

Exemplary embodiments are thus directed to budget constrained biddingstrategies in keyword auctions. Embodiments include a single keywordcase and strategies for both single-slot and multiple-slot cases. In thecase of single-slot auctions, one embodiment provides obliviousstrategies for both profit and revenue maximizations. The obliviousnessis a desirable property in any bidding strategy. In the multiple-slotcase, one embodiment has a strategy for profit maximization wherein itis no longer oblivious and requires knowledge of the various bids andclick-through-rates of slots. However, for revenue maximization, itstill remains oblivious.

Exemplary embodiments can be extended to the general case where thereare multiple keywords and each keyword has multiple positions. Thecompetitive ratio would now have V replaced by V_(max), where V_(max) isthe maximum valuation for all keywords.

In one exemplary embodiment, one or more blocks in the flow diagrams areautomated. In other words, apparatus, systems, and methods occurautomatically. As used herein, the terms “automated” or “automatically”(and like variations thereof) mean controlled operation of an apparatus,system, and/or process using computers and/or mechanical/electricaldevices without the necessity of human intervention, observation, effortand/or decision.

The flow diagrams in accordance with exemplary embodiments are providedas examples and should not be construed to limit other embodimentswithin the scope of embodiments. For instance, the blocks should not beconstrued as steps that must proceed in a particular order. Additionalblocks/steps may be added, some blocks/steps removed, or the order ofthe blocks/steps altered and still be within the scope of the invention.Further, blocks within different figures can be added to or exchangedwith other blocks in other figures. Further yet, specific numerical datavalues (such as specific quantities, numbers, categories, etc.) or otherspecific information should be interpreted as illustrative fordiscussing exemplary embodiments. Such specific information is notprovided to limit the exemplary embodiments.

Various exemplary embodiments are implemented as one or more computersoftware programs. The software is implemented as one or more modules(also referred to as code subroutines, or “objects” in object-orientedprogramming). The location of the software (whether on the clientcomputer or elsewhere) will differ for the various alternativeembodiments. The software programming code, for example, can be accessedby the processor of the computing device 20 and computer system 40 fromlong-term storage media of some type, such as a CD-ROM drive or harddrive. The software programming code can be embodied or stored on any ofa variety of known media for use with a data processing system or in anymemory device such as semiconductor, magnetic and optical devices,including a disk, hard drive, CD-ROM, ROM, etc. The code can bedistributed on such media or can be distributed to users from the memoryor storage of one computer system over a network of some type to othercomputer systems for use by users of such other systems. Alternatively,the programming code can be embodied in the memory and accessed by theprocessor using the bus. The techniques and methods for embodyingsoftware programming code in memory, on physical media, and/ordistributing software code via networks are well known and will not befurther discussed herein.

The above discussion is meant to be illustrative of the principles andvarious exemplary embodiments. Numerous variations and modificationswill become apparent to those skilled in the art once the abovedisclosure is fully appreciated. It is intended that the followingclaims be interpreted to embrace all such variations and modifications.

1) A method, comprising: receiving bids for advertising at an onlinesearch auction; bidding for the advertising an amount equal to anexpected value-per-click divided by a value of a function of a fractionof a budget already spent for advertising on previous bids at the onlinesearch auction; displaying advertisements in winning ad slots. 2) Themethod of claim 1 further comprising, placing a bid only when a highestbid among current bidders for the advertising is less than the expectedvalue-per-click. 3) The method of claim 1, wherein an efficiency of eachad slot has an upper bound based on the expected value-per-click dividedby a search engine minimum fee charge for receiving a slot to advertise.4) The method of claim 1 further comprising, calculating an optimalamount of money to bid for the advertising based on a multiple-choiceknapsack problem modeling of ad slots over time periods. 5) The methodof claim 1 further comprising, modeling of a multiple-choice knapsackproblem based on one of maximizing a total revenue of an advertiser overtime, maximizing a total profit of the advertiser, or a total number ofimpressions. 6) The method of claim 1 further comprising: modeling anonline trading process of goods or services, wherein a trader has abudget constraint as an online knapsack problem; solving a onlinetrading problem using an algorithm developed for the online knapsackproblem. 7) The method of claim 1 further comprising, determining anoptimal amount of money to bid for the advertising slot without havingknowledge of other bids and click-through rates for advertising slots.8) The method of claim 1 further comprising: receiving keywords for asearch query; assessing the keywords to determine how much to bid for anadvertising slot in order to maximize a return on investment for thesearch query. 9) A computer-readable medium having computer-readableprogram code embodied therein for causing a computer system to perform:receiving bids for advertising slots for a network search query; biddingfor one of the advertising slots an amount of money that is an expectedvalue-per-click divided by a function of a fraction of a budget alreadyspent for previous advertising slots; displaying advertisements ofbidders. 10) The computer-readable medium of claim 9, wherein the codefurther causes the computer system to perform: calculating when amaximum bid amount for the advertising slots is greater than theexpected value-per-click. 11) The computer-readable medium of claim 9,wherein the code further causes the computer system to perform:determining the amount of money to bid based on profit for winning theone of the advertising slots divided by a cost to win the one of theadvertising slots. 12) The computer-readable medium of claim 9, whereinthe amount of money depends only on a value of keywords submitted forthe network search query and the fraction of the budget already spent.13) The computer-readable medium of claim 9, wherein the code furthercauses the computer system to perform: determining an optimal amount tobid for the one of the advertising slots without utilizing informationpertaining to (1) amounts of other bids received for the one of theadvertising slots and (2) how frequently network search queries occur.14) The computer-readable medium of claim 9, wherein each of theadvertising slots has a click-through rate that is defined as anexpected number of clicks on an advertisement divided by a total numberof impressions. 15) A computer system, comprising: memory storing analgorithm; processor to execute the algorithm to: examine bids foradvertising slots for a keyword search; submit a bid amount for theadvertising slots, the bid amount being a function of an expectedvalue-per-click divided by a value of an exponential function of afraction of a budget already spent; allocate the advertising slots tobidders. 16) The computer system of claim 15, wherein the bid amountdepends only on a value of keywords for the keyword search and thetraction of the budget already spent. 17) The computer system of claim15 wherein the processor further executes the algorithm to: place a bidonly when a highest bid among current bidders for the advertising slotsare less than the expected value-per-click. 18) The computer system ofclaim 15 wherein the processor further executes the algorithm to:calculate an optimal amount of money to bid for one of the advertisingslots without utilizing information of other bids and click-throughrates for the advertising slots. 19) The computer system of claim 15wherein the processor further executes the algorithm to: calculate anamount to bid for one of the advertising slots in order to maximize areturn on money budgeted for advertising at online auctions. 20) Thecomputer system of claim 15 wherein the processor further executes thealgorithm to: calculate an amount of money to bid based on profit forwinning one of the advertising slots divided by a cost to win the one ofthe advertising slots. 21) The computer system of claim 15 wherein theprocessor further executes the algorithm to: calculate an optimal amountto bid for the advertising slots without utilizing informationpertaining to a frequency of how often search queries are executed. 22)The computer system of claim 15, wherein each of the advertising slotshas a click-through rate defined as an expected number of clicks on anad divided by a total number of displays.